Editing Radial velocity (section)

==Formulation==

Given a differentiable vector <math>\mathbf r \in \mathbb{R}^3</math> defining the instantaneous [[relative position]] of a target with respect to an observer.

Let the instantaneous [[relative velocity]] of the target with respect to the observer be

{{NumBlk|:|<math> \mathbf v = \frac{d\mathbf r}{dt} \in \mathbb{R}^3</math>|{{EquationRef|1}}}}

The magnitude of the position vector <math>\mathbf r</math> is defined as in terms of the [[inner product]]

{{NumBlk|:|<math>r= \|\mathbf r\| = \langle \mathbf r,\mathbf r \rangle^{1/2}</math>|{{EquationRef|2}}}}

The quantity range rate is the [[time derivative]] of the magnitude ([[Norm (mathematics)|norm]]) of <math>\mathbf r</math>, expressed as

{{NumBlk|:|<math>\dot{r}=\frac{d r}{dt}</math>|{{EquationRef|3}}}}

Substituting ({{EquationNote|2}}) into ({{EquationNote|3}})

: <math>\dot{r} = \frac{d \langle \mathbf r,\mathbf r \rangle^{1/2} }{dt}</math>

Evaluating the derivative of the right-hand-side by the [[chain rule]]

: <math>\dot{r} = \frac{1}{2}  \frac{d \langle \mathbf r,\mathbf r \rangle}{dt} \frac{1}{r}</math>

: <math>\dot{r} = \frac{1}{2}  \frac{\langle \frac{d\mathbf r}{dt}, \mathbf r \rangle + \langle \mathbf r,\frac{d\mathbf r}{dt} \rangle}{r}</math>

using ({{EquationNote|1}}) the expression becomes

: <math>\dot{r} = \frac{1}{2}  \frac{\langle \mathbf v,\mathbf r \rangle + \langle \mathbf r,\mathbf v \rangle}{r}</math>

By reciprocity,<ref>{{cite book| last1=Hoffman|first1=Kenneth M.| last2=Kunzel|first2=Ray| year=1971| title=Linear Algebra| edition=Second| publisher=Prentice-Hall Inc.|page=[https://archive.org/details/linearalgebra00hoff_0/page/271 271]| isbn=0135367972|url-access=registration| url=https://archive.org/details/linearalgebra00hoff_0/page/271}}</ref> <math>\langle \mathbf v,\mathbf r \rangle = \langle \mathbf r,\mathbf v \rangle</math>.
Defining the [[unit vector|unit]] relative position vector <math>\hat{r} = \mathbf r/{r} </math> (or LOS direction), 
the range rate is simply expressed as

: <math>\dot{r} = \frac{\langle \mathbf r,\mathbf v \rangle}{r} = \langle \hat{r},\mathbf v \rangle</math>

i.e., the projection of the relative velocity vector onto the LOS direction.

Further defining the velocity direction <math>\hat{v} =\mathbf v/{v} </math>, with 
the [[relative speed]] <math>v =\|\mathbf v\|</math>, we have:
: <math>\dot{r} = \langle \hat{r},v\hat{v} \rangle = v \langle \hat{r},\hat{v} \rangle</math>
where the inner product is either +1 or -1, for parallel and [[antiparallel vector]]s, respectively.


A singularity exists for coincident observer target, i.e., <math>r = 0</math>; in this case, range rate is undefined.